^c 

CC 

CC < 
<r<c 

cr 4 


^^ < 


1 




^B 


^ 


1 


3E 




1 


cj 




a 


c 


cr « 


rfl 






4 


< 


^< < 


^B 



<£Tcc 






-c 5F C< ^ 

-> *gL cc; 
c *£! cc 

^ *C^ cc 

S-^CC 

"^ CC 



cc 

cc 



c c 

=i c 
JC C 

.5 c: . 



J LIBRARY OF CONGRESS .t 

i 

0§Jan 




opsrijW |[a; 




# 



I UNITED STATES OF AMERICA.? 



C«M"<r< 



^' cr 



- c 
c 



CC 

cc 
cc 

cr 

cc 

cc 
s cc 

cc 

cc 
~ cc 

_ cc 
cc 

cc 

Cr 
cc 



^.<t 



- cc 

.CC 
O; <^ 

CCc 



__ Cc 
:<cc; 

sC2t 






*>>« <ac:" 



CC ^ 

: ^*R "cc 

rcc 



ccc 



■■<> 

i cr 



Cc 
^C^L. <g 



.ccc 



<T c<; 
Ccc 






c 


ccr ^c^ic<^. <&c«£ 


... 


c 


x:<£* «3£lc<<c: <x<:<. .<< 


'Z~ SlSS 


<: 


:«c«c< 7 ;:*C£j<c«: <tm^m 


S- c 


C 


C<S '^K5t<C^<S*gMI 


;.<. -■ ! 


5 < 


c<r • ; «s«j<c: <yc^ « 


^^ '■- 


c 


c<i ^^csci : -'<§«a3ii 


IT - " ' ! 


c 


c<i ^K3cc«i;: csfec^i 


CZ1 i 


^ 


cxzi oc^cd <sgg^m 


^ f C ' 


<I 


CC *££.*« . : <2SE§^«H 


M, ~ ■"<■ 


;c 


: occ «zcc«^: <s<3^:m 


Jf j ' o 


""IT 


</</ 4C^^C:.<SSCJC^ 


czzi" 


2T_ 


<C_<Ij ( ' C-'<C <" ^ 


KL <i: . 


^ ( 


CI<3 C </<3c « 


8ZZH ^"' 


£_ 


cz_<~c. . *et<«c c «c ' c ... « 


tZ/ ■-' 




<z:<r_5. «^.c«r-. <L<cr*:" < 


^KZI 



r"<£c «c~« 
'fee c <l c_ 



.5SC CI. "<E <*< 

c cc<: 






_ c?cc < 
-c: c - 

_ C <L 

/CO 






< € <3T 
€ C 

~«3CC •< . 



< <? 



2-ccIc" «d ■■■■' 
Co <Zc est •.-<-■■ 



<LI <T<?:. c: 



^<-^Tjfc 









: </•<</ c 

3SKZ.CT. C : <C^<rr^.< 









<L~~<1I t 



zr-<« 



1 d '<3<"c - CC 

<T' xr^ ; - c - <x ere < 

«r." <?- •■ 

r «: <s. creche 

; «c <s cc 



THE CELEBRATED 

THEORY OF PARALLELS. 



iDEnvcojsrsT^i^^Tioisr 

OF 

THE CELEBRATED THEOREM. 

EUCLID L AXIOM 12. 



With Appendix containing the philosophy of the demonstration, together with the partial 

refutation of Sir Wm Hamilton's philosophy of the 

Unconditioned or Infinite. 



OF COUNTY TTPPERARY, IRELAND, 

Clerk. War Department, (Office of Accounts, Gen. Chauncey McKeever.) Washington, D. 0. 

[Late 3d Reg"t U. S. Infantry.] 



But you forget that Geometrical Equality can do great things, both among pods and men," 

— Plato. Gorgias- 




Thus, also, universally, from the comparison of the equality of finites may be evolved some positive knowl- 
edge of the corresponding homogeneous Infinites, whether in Deity, space, time or degree. 

—Appendix, Note A, Par. c. 



WASHINGTON, D. C: 
CHRONICLE PRINT 
1866. 



Washington, D. 0., ( War DepL, Office of Accounts,) 

January, 1866. 
SIR: 

This work, which the author takes pleasure in presenting. 
besides wiping away the " reproach of Geometry" defeat s 
atheism and infidelity. {See foot note, page 3: also, 'note B, 
Appendix,) 

Positive ideas of infinity other than parallelism are estab- 
lished in Lemmas II and III of the manuscript referred to in 
foot note, page 4. 

N. B. — Acknowledgment of receipt hereof; also, a copy of 
your review of the work, will oblige 

Yours, very truly, 

Matthew By ax. 



THE CELEBRATED 

THEORY OF PARALLELS 



IDEIs^OlsrSTDEi^^TIOlsr 

OP 

THE CELEBRATED THEOREM. 

EUCLID I, AXIOM 12. 



With Appendix containing the philosophy of the demonstration, together with the partial 

refutation of Sir Wm Hamilton's philosophy of the 

Unconditioned or Infinite. 



V 

IBY MATTHEW Tt^ZTJ^TST, 

x ' . OF COUNTY TrPPEKARY, IKK LAND, 

Clerk, War Department, (Otfice of Accounts, Gen. Cha.uncey McKeever,) Washington. D. C. 

Late 3d Reg't U. S. Infantry.] 



" But you forget that Geometrical Equality can do great things, both among gods and men." 

—Plato. Gorgias- 




Thus, also, universally, from the comparison of the equality of finites may be evolved some positive knowl- 
edge of the corresponding homogeneous Infinites, whether in Deity, space, time or degree. 

—Appendix, Note A, Par, c. 

IF 



WASHINGTON, D. C: 

CHRONICLE PRINT 
1866. 



-^4- 



Entered according to Act of Congress, in the year 1S65. 

T 

BY MATTHEW RYAJS T , 
In the Clerk's Office of the District Court of the United States for the District of Columbia. 




SKET C H 



OF THE 



Romantic History of i arallels. 



(a. ) Of all contested geometrical subjects that of Parallels — the great blem- 
ish in the immortal " Elements" of Euclid — is the most ancient as well as the 
most fascinating. At the porch of science and clasping infinite Space, the sub- 
ject is at once simple and sublime. Its recorded history commences with the 
mild and the benevolent Euclid, 280 years B. C. (see Euc. I, 28 ;) and 
though forming an essential element in most demonstrations, yet, as an un- 
established theory, it must have been handed down by the illustrious Pytha- 
goras, by the venerable Thales, and probably by the Egyptian priests who 
were the instructors of these philosophers. 

(b.) Says Larclner, "The Theory of Parallels has always been considered 
as the reproach of Geometry.* # »" In pamphlet published 185G, 

T. P. Thompson f declares, '• # # # Ptolomey, Proclus, 

the Arab editor of Euclid, Clavius, Wolfius, Boscovich, D'Alembert, Thomas 
Simpson, Bonnycastle, Robert Simson, Varignon, Bezout, Leslie, Ludlam, 
Playfair, Franceschini, Legenclre, La Grange, Le Croix, Bertrancl, with 
others of later date have successively set their seals to the admission that 
till something was done on this pointd: Geometry was not entitled to the name 
of an exact science." 

(c.) Stevelly, (Lond. Edin. and Dub. Phil. Mag. 1856, Vol. XII, p 220,) 
frankly acknowledges thus: "I was again induced to waste some hours on 
a subject on which in my schoolboy and college days I, in common I sup- 
pose with every schoolboy and collegian since the days of Euclid, had over 
and over again wearied myself in vain. ' ' 

(d.) During the past half century the matter had been subjected to the 
most subtle analysis ; the doctrine of functions, and the inadmissible doctrine 
of limits had been appealed to, but in vain. In the investigation of the 
subject, indeed, " many have lost much time, and some even their reason. * 
* *."§ Its investigations have embittered the most exalted friendships. In 
this research the philosophic minds of Leslie and Legendre have waged war. 
Finally many moderns concluded that "to deduce the property of two line" 
postulated as parallels involves a direct dealing with the positive idea of infin- 
ity — a task utterly beyond the reach of our faculties," &c. || Similar, but 
more elaborate declarations will be found in an article on parallels by James 
Adamson, D. D., in the London, Edinburgh and Dublin Phil. Mag., 1853, 
Vol. V, page 407 ; also in the Penny Cyclopaedia, Vol. XVII, art. paralleU > 
concluding paragraph, page 238, as also in other works. Thus at length 
did this mysterious subject powerfully aid in the founding of a false and 
atheisticalphilosophy,^[ and bid fair to remain a matter of contention during 
all the ages of the future. But God has been pleased to hearken to the voice 

*Dionysius Lardner, LL.D.— P. R. S — L. & E. (Geo. 1S38.) 

fGeneral T. Perronet Thompson, of Eliot Vale, Blaekheath, London. 

^Meaning parallels. 

(^Encyclopedia Americana. — Subject, Parallels. 

||Lond. Edin. and Dub. Phil. Mag. 1S57, Vol. XIII. page 413, as quoted from " Nichols' Cyclo- 
paedia of the Physical Sciences." 

U'Thus, on the hypothesis that a positive idea of the Infinite is impossible to the Iranian mind, 
false philosophers have contended that hence tha existence of an Infinite Being is unsuscepti- 
ble of demonstration, and is therefore a superfluity, and as such, an evil as an element of man's 
religious belief. 



of the illustrious dead, and to reveal the hiatus of twenty ages as the reward 
of eight years of an Irish exile's diligent investigation. ("See prop. B.)* 

(e7) It is curious to observe that the hist : ry subsequent to the demonstra- 
tion of prop. B. is no less romantic than that which precedes. Thus, in 1860, 
in order to bring this demonstration before the world, the author visited 
France, but in vain, owing to the mathematician Bertrand's absurd (an " 
the author conceived, proud) demand of the custody of the manuscript. In 
June, 1864, the American Institute of New York refused to pronounce upon 
the demonstration. In December, 1864, the demonstration was deposited 
with Prof. Joseph Henry, of the 8n ' vian Insii. rton. D. C, 

who unphilosophically shrank from declaring an opinion, though twice en- 
ted by the writer so to do. Prof. Henry finally announce 1 the loss of 
the manuscript in the conflagration of February, 1S65. Prof. HDdgai " 
the Coast Surrey, and others, have also shrank from expressing oj 
But peace must be given to the Divine Science, and Philc jphers must not 
be debarred from the delight of knowing a beautiful truth through the jeal- 
ousy, the unphilosophical timidity, or the prejudice of Bertrand, of Henry, or 
others : and therefore the Author publishes, at his own expense, (and out of 
a modest salary,) his establishment of the TJieory of Parallels, and freely 
distributes the same throughout every land.f 

(f. ) The Author tenders his love to the Mathematicians and Philosophers 
of every nation, and in so doing confesses his conception that in Append 
Note B, is pointed out an interesting field of geometric and philosophic 
thought ; thus giving an impetus to man's approach to that Ultimate of all 
Truth — the contemplation of the unveiled glory of that Lntlxitl I:r._: 
gexce — that Cause or all — the Eternal Uncaused. 

MATTHEW RYAN. 
"War Department, (Office of Accounts,) 
Washington, D. C, Decern; - " 

*For fall information see "J ypcei Vol. XV 11 E . " '-Loud. 

Edin. and Dub. Phil. Mat) -" T.P.Thc . ;'s Geo., 1S34: Prof. Jas. Thomp- 

ton's Euc.\ also. Car.-: : _ . . E srlin. 1S2% kc. 

fTMs pamphlet is but a fragment of the manuscript work, " Tie Perfect Geometry,"" icwMch 
contains the theories of the straight line and plane; also an Appendix containing :"_r refutation 
of certain sophisms and n. : . hysical ": :: :ns brought against prop. A, some original props., 
some positive knowledge of the Deity, &c. Academies desiring to publish may apply for 
manuscript. 



DEF. A. (GEOMETRICAL COINCIDENCE.) 

Coincidence is the term given to the property of capability of exact occupa- 
tion of the same space possessed by homogeneous and finite magnitudes, if 
superposed. 

Cor. Coincidence assumes five forms, of which four are direct and one in- 
direct. 

1 — The immediate form, as in the case of identical magnitudes. (Euc. I, 2, 
4, 5, 8, 26, &c.) 

2 — The transposition form, as in dissimilar figures. (Euc. I, 35, 36, 37, 38, 
42, 43, 44, 45 and 47 ; Euc. II, 11 and 14 ; and Euc. VI, 31, &c.) 

3 — The repeated immediate form, as where the several parts of one magni- 
tude coincide at once respectively with the several parts of another. 

4 — The continuous repeated immediate form, as in the superposition of the 
circumference on the straight line, &c. (See prop. A.) 

5 — The indirect form, or form of comparison with equal auxiliaries, or form 
wherein the magnitude of each thing is determined by identical conditions, as in 
the case of A A / , B B / , fig. prop. A,* &c. 

Schol. — Since a theorem is established (as all logicians admit) when its 
converse involves an impossible consequence ; therefore, the certainty of co- 
incidence in any case is established by inferring an impossibility from the 
contrary assumption. 

N. B. — This simple schol. is the bulwark of prop. A. 



DEF. B. (GEOMETRICAL EQUALITY.) 

The capability of coincidence is called equality, f 

Cor. 1. Equality can alone be predicated of finite magnitudes. [See Def. 
A.]J 

Cor. 2. (a.) The ideas of equality and coincidence are involved in one 
idea, " for equality is nothing but the capability of coincidence ;"§ and since 
the idea of coincidence involves the idea of simple motion ; therefore also the 
conception of equality involves the conception of simple motion. The same 

*Thus is coincidence subject to that grand law of variety which is observable throughout all 
the operations of nature, and all the departments of knowledge, save alone the immutable laws of 
Geometry. 

t "* * * from whence it is sufficiently apparent that by Equality is understood nothing 
else but & possible Congi-nity." Barrow's Lectures No. XI. 

£*' The observation that equality (meaning of magnitude not of ratio) implies being finite, is as 
old as the Philebus of Plato. ' And next, all the circumstances that appeared to be incompatible 
with it ; as first, having equal or equality ; and after that, having a double, or anything else that 
implies relation of number to number or measure to measure; all these we set down as appro- 
priate to the finite only.' — Plato, Philebus. Plato's distinction will be found exact if limited to 
what he manifestly had in view. No man can attach any rational idea to half eternity ; nor, by 
analogy, to twice eternity. If there are to be two co-existing eternities, (as for instance two rec- 
tangular parallelograms on contiguous bases, whose altitude is of unlimited length,) this is a 
question of ratio; which is quite another thing." T. P. Thompson's Geo. oth edit. p. 132. Again, 
at page 153, "All references to the equality of magnitudes of infinite surfaces, in respect to the 
parts where they are avowedly without boundaries, are intrisically parallogisms ; for it is tan- 
tamount to saying that boundaries coincide where boundaries are none." [This last is an imme- 
diate refutation of M. Bertrand's attempt to establish the properties of parallels. Author.] 

§T, P. Thompson's Geo., oth edit., 1834, page 148. 
5 



may be said ol meqmiiUuv. riz.: that its conception inYolres the conception of 
coincidence ; that is, of partial coincidence, and theiefoie of simple motion,. 

for iaegK&IsSf m smply tne eqmaHtig ef apart. 

";. JT;:v-: .- . _ ; r _\: ZZz- ±:~ iZz Zz-_zz :: Zziz -_ I: : "_: ■/" .- 
- : - =.--.-"-.-'-• 7 :."_"'_:::'_: e z Z : z. : : . z : _ _ _ - : : ~_ : :- z Z " "_ : : 7 :,;:.... -: . 5 Z_~ z-3 
= ---- -_• — : ; " ; z-2 z : : : :- .: '. -. 1 ~. - '-. : : - : - z~ z : ~ : 11 : " - : z 
_ : Z - ,-.- 11 I in~:I~fi il: z:~ zz-tZiz: Z~:Z:z 



: z :n: 



: . ::z._ z 



__z::::^ 



Z_ _ — .£ 

-"' " -■ ~ ■- L ' '- 
_ ZZ — _ 

.5ei#L From these two 

iZz I -:.'~ '..-.-. z~ Z~f 1 zZ 



. - : : Z : • Z ■ -. : z Z 



_ ■ :.\ rzsiiz: z a ix— : _._:: 

; :r: Z ; Z __ rizZzi Z : :Zz zZZ zizzZj ::' zZli zZn :zz5ZZ"z.z: : I 
Z ;■"■:. Z " : Z. I • " Z, : ' ■ : : zzi ; : : ' Z - Z \ :■: zz :.'..:: . 

Zz !..;^puv;':' z ;Zz -, Z z.;z~ I Z . zzZ «z Z Z.z; Z jZ;Z :'zzj 
a gmem imdmatiam t» $ka& atiler, ami, (2) fiat £1$ same pevmt (B) t a «f afea 
alnegs ?m fa £&a? ether; IhmskaM Urn Uca* (A AO afammpamt (A> «tlea« 
; Zz.Z: z.;'-' '": ' : " I Z " : zz'Z z z ■ - Z' ' " • - 

Gasel. Let A B be perpendicular to B C. Bisect A B in D, draw Z Z- 
.r.Z :: Z Z :Z: __ z.z: I 7 = Z Z _. : _ Z Z=zr: _ :.. .: I I i.T. 
Zz: Z 7. :- ;:i:t:- — _ ■. Z I _-_ 7 :z--Z-z z:zi I :zz :Zt .-■ z_ : _ . z 
z.zz-Z:r:\"z.::"L-:::z: — : 1 _\: :z "z Z /_ Z i: _ : :: I 
ZzttZzz. zzt-zzzz :~ :: I Z U^S- 1-) 

: : : 1 - _L 8f 



:.; 



\ : : _ _ -" - 
:-:~:Z z:z: .z Z 




1 

;: ' ; . ^ ' -- - : 'r "_ ; : '_ _ :' : 

•*'-■ ::. -_:-~ ~ . 1.:.:: :'_.r : -: 
-Z — ._.. -;Z :_^ Z_i : .:' 



- 



7- : • : Ti.:'; :•::-.?■? ._--.- : : 1: : • ~ li: : :: - —.'.'. 



For, conceive 
{Mg. 2.) 




(a.) No two points in B E pass through any one in B C. 
that a, b, ("fig. 2) in B" E" pass through B / 
in B C. ^hen, because the arc a b must 
pass through B / , whilst the moving line re- 
mained in position A / B / ; therefore B / would 
not have moved on B C equally with arc B 7/ 
E" through B'; which (Hyp.) is impossible. 
Wherefore the assumption is false. Neither 
can two points c, B / in B C (fig. 2) move 
through any one point W in arc B // E /7 ; 
for then likewise, B C and arc B E ("fig. 1) 
would not have moved equally through B. 

(b.) No two points in A F (fig. 1) pass through any one in A A / . Sup- 
pose that d, e, (fig. 2) pass through A / in A A / . Then, since the arc d e 
must pass through A 7 , draw d a : and because d e passes through A 7 , there- 
fore a b, which is equal to d e, passes through B', which (Paragraph a) is 
impossible. Neither can any two points in A A / ffig. 1) move through any 
one in A F. For, suppose that f and A 7 , (fig. 2) and hence line f A / pass 
through A / in arc A // F // ; therefore the corresponding portion c B' on B B / 
passes through W in arc B" E", which (Par. a) is impossible. 

(c.) Now, the function performed by B E on B B 7 , and by A F on A A', 
is that of exact coincidence, or the establishment of the equality of B E, 
B B', and of A F, A A / (fig. I). For, the denial of the equality of B E, B B', 
is "simply the denial of the capability of coincidence of B E, B B'. Assum- 
ing, therefore, that a part of B E, as a b, is incapable of coincidence with 
B B 7 , it follows that a b moves through a single point in B W, which (Par. a) 
is impossible. Similarly is it proved impossible for a part c d of B W to be 
incapable of coincidence with B E. "Wherefore, every element of B E coin- 
cides with an equal and corresponding element of B B', and therefore [Def. 
A, Schol.) BE,BB / coincide throughout, and wherefore (Def. B) B E=B B'. 

(d.) Also, arc A F=A A / (fig. 1). For, the denial of this is simply the 
denial of the capability of coincidence of A F, A A / . Let e f therefore be 
assumed a part of A F that is incapable of coincidence with A A'; and it fol- 
lows that e f passes through a single point in A A 7 , which (Par. b) is impos- 
sible. Similarly is it proved impossible for a part g h of A A / to be incapa- 
ble of coincidence with A F ; therefore every element of A F coincides with 
an equal and corresponding element of A A', and wherefore (Def. A, Schol.) 
A F, A A / coincide throughout ; and therefore (Def. B) A F=A A / . 

(e.) Now, since arc A F=A A / , and arc B E=B B'; also, since arc A F 
=arc BE,* therefore A A 7 =B B / . Similarly is it proved that if A B move 
on B C for other lengths equal each to B B / , then also the lines traced by A 
are equal respectively to the lengths described on B by B : and therefore 
the points A, B describe equal lengths. 

Case 2. — Let A B be oblique to B C; draw A D perpendicular to B C, and 
conceive triangle A B D to move upon B C in 
the same plane. Let A A / be the locus of its 
vertex, and B B', D D' the lengths described by 
either extremity of the base. Then (Case 1st) 
A A / =D D'; but D D / =B B'; therefore A A' 

=B B . *The proof that vertical arcs are equal is simple.. 




8 



Lemma. — In either of two intersecting straight lines (A B, A C), or continu- 
ation of either ; to find a point (E), such that any line between it and the other 
line shall exceed any given line (D).* 

Let B A C be acute, and let n denote the times B A C is contained com- 
pletely in 4 right angles. Take A E=J (Dxw) : from A with radius A E 
describe circle cutting A C in F, H ; draw E K perpendicular to A F, and let 
E K cut circle in G, and join A G. In triangle 
AEG (Euc. I, 5) angle A E G=A G E : then in 
triangles A E K, A G K, the angles A E K, A K E 
=A G K, A K G, each to each ; and A E=A G ; 
therefore (Euc. I, 26) angle E A K=G A K ; and 
wherefore angle E A G=2 E A K. Make angle 
G A L=E A G : if L lie between H and E, join 
G L, L E, H G : and since E G, G L envelope E A, 
A L, therefore (Euc. I, 21) E G+G L>E A+AL, 
or 2 A E. But E G or G L=2 E K : also angle E A G+G A L are not 
greater than BACXw; and therefore E G+G L are not greater than EKx 
n ; and wherefore E Kxw>2 A E or Dxw: therefore E K>D. If angle 
B A C diminish so that L falls between H and G, then Euc. I, 24, is quoted in- 
stead of Euc. I, 26. The solution of the case wherein B A C is obtuse is in- 
volved in the foregoing. The solution of the case wherein B A C is a right 
angle is effected simply by taking A E greater than D. 




D 



PROPOSITION B THEOREM. 
[Being the celebrated so-called 12th Axiom of Euclid.] 

If a straight line (A C) meet two other straight lines (A B, C D) which are 
in the same plane, so as to make the two interior angles (B A C, A C D) on the 
same side of it, taken together, less than two right angles, these two straight lines 
(A B, C D) shall at length meet upon that side, if sufficiently produced. 

Draw A E making angle E A C+A C D=2 right angles. In A E or its 
production find {Lemma) F, such that any line between F and A B shall ex- 
ceed A C. Produce A C to G. Let A F move on A C as enunciated in prop. 
A, and until F falls on A B or its production, as F'; 
where A' W is the new position of A F, and F F / the 
locus of F. Because Cconst. ) F F'> A C ; and because 
(prop. A) F F / =A A'; therefore A A'>A C. Since 
angle E A C+A C D=2 right angles ; and since an- 
gle A C D+D C A / =2 right angles, therefore angle 
D C G=E A C=F' A / G ; and wherefore C D, A F / 
are parallel; for, if C D, A / F / meet towards D, 
F', then (Euc. I, 16) angle F' A' G>D C A'; but * 
this is not so. Neither can C D, A / F / meet towards C, A', for then (Euc. 
I, 16) angle vertical to D C A / would exceed angle vertical to F / A / G ; but 
it does not. Therefore C D, A' F / are parallel : and since A / F / meets A B 




*This Lemma was assumed by Proclus ; also by Aristotle in his attempt 
the world is finite." 



'to establish that 



9 

produced, C D shall likewise meet A B, or production of A B within the 
figure F A A 7 F / : which was to he demonstrated. Therefore, " If a straight 
line meet," &c. 

APPENDIX.— NOTE A. 

On the beautiful philosophical relation between the mode of proof in prop % A 
and the establishment of the knowledge of Parallelism. 

(a.) Coincidence, or the capability of coincidence, and known as equality , 
is the basis of all geometric knowledge. Equality is, in all cases save one, 
manifested in the direct manner ; that is, in either of the first four forms set 
forth in cor. to Def. A. Thus, by direct coincidence is established the equal- 
ity or inequality of the sides, angles or areas of the two magnitudes in Euc. 
I, 4, 5, 6, 8, 18, 19, 24, 25, 26, 85, &c, to 47 : also Euc. Ill, 20, 21, 24, 28, &c; 
as well as the knowledge of all propositions which ultimately rest on Euc. I, 
4, as basis. In these all, also, the idea is composed of two sub-ideas or terms, 
which are unrelated except by comparison, or by possessing the property of 
equality. Alone Parallelism presents a single conception — consists of but one 
term, that of non-concurrence. Other subjects present to the mind two homo- 
geneous magnitudes and their relations ; and when the subject is the equality 
or inequality of two lines or angles, these are related with surfaces or solids. 
In other subjects the datum is equality or inequality, or their combination 5 
but in parallelism the datum simply is that the lines shall never meet. 

(b.) To discover new immutable truths, the human mind requires at least 
two truths to work with (vide Elucid. to Ax.): and since parallelism commands 
but a single conception — that of non-concurrence — therefore in the establish- 
ment of the laws of parallels, the 4 direct forms of coincidence yield to the 
5th or indirect form (see Def. A); and hence, two identical auxiliaries must be 
used, and be simultaneously superposed on either parallel. Such auxiliaries 
are alone found in vertical arcs of the circumference. Thus, the clasping of 
either parallel by the equal arcs A F, B E, and the simultaneous coincidence 
of these arcs with A W, B W, fulfill the functions of the essential principle 
of direct coincidence. 

("a ) The identity of the circumference in all its parts compels the eternal con- 
tinuance of any law established by the least rotation of the circumference. Thus 
is the circumference the unique symbol of that infinity which is embraced in 
Parallelism.* By no properties of straight lines alone could the knowledge 
of parallels be established, for the straight line involves only finite considera- 
tions, and the direct forms of coincidence. Whilst, also, on the one hand, 
the circumference in its form contains a certain infinite laio, and in its being a 
finite lohole, (and therefore every arc a portion of a finite whole) lies within 
the comprehension of the human mind ; on the other hand, every straight 
line is but a portion of an infinite whole, and hence, of itself, fails to present to 
the mind those certain infinite laws which have a peculiar existence in par- 
allelism, f Thus, by the consideration of the equality of finites, viz. : B E, 

*The demonstrative knowledge of many properties of the circle is still veiled from man. Some 
of these occur in the revolution of a radius about the centre. 

fWe can therefore account for the failures of those forty and more ancient and modern attempts 
wherein was alone employed the straight line. T. Perronet Thompson, alone of all Geometers, 
appealed to the circle, but in vain, because not having employed the circumference in that auxil- 
iary method given in prop. A. SeeT. P. Thompson's Pamphlet, 1S56. 



10 

B B / , (fig. prop. A) do we establish a knowledge of an Infinite, viz. : Parallel- 
ism. Thus, also, universally, from the comparison of the equality of finite* 
may be evolved some positive knowledge of the corresponding homeogeneous 
Infinites, whether in Deity, space, time or degree. 

(d.) A remarkable sophism or parallogism committed by a certain member 
of the American Institute of New York City is worthy of record. This gen- 
tleman denied the capability of coincidence of a curve and straight line (see 
B E, B W, fig. prop. A) either in whole or part, but admitted their equality. 
This sophism or parallogism is refuted by Ax. I and II. Thus (Ax. I) B E, 
B B / , are either equal or unequal, but (Ax. II) not both. If the equality of 
B E, B B / be admitted, (Def. B) the capability of coincidence of B E, B B' 
is thereby admitted ; whilst if the inequality of B E, B B / be admitted, from 
the very signification of inequality (Def. B, cor. 2, par. a) B E, B W if 
superposed, would coincide in part. Thus, the vague principles of Meta- 
physics fall before the two axioms of Geometry. 

NOTE B. 

On the establishment of a positive knowledge of the Unconditioned or Infinite 
in Space. Being a partial refutation of Sir Wm. Hamilton' 's philosophy of the 
Unconditioned or Infinite. 

(a.) The knowledge established in prop. B is positive because demonstra- 
tive; it is entire, and is that of an Infinite Whole, for parallelism involves In- 
finite Space both in the direction of the lines and their distance apart. Such 
is admitted even by the Hamiltonians. Saith a disciple of Hamilton, "to 
deduce the properties of two lines postulated as parallels involves a direct 
dealing with the positive idea of infinity — a task utterly beyond the reach of 
our faculties."- Sir William, himself, conceives that " The unconditioned is 
incognizable and inconceivable;" that "the mind can conceive, and con- 
sequently, can know, only the limited and the conditionally limited," &c, 
and that "The result is the same, whether we apply the process to limita- 
tion in space, in time, or in degree." f 

(b.) Hamilton, whilst refuting the German and French metaphysicians 
who impiously ascribed a complete knowledge of the Infinite or Absolute to 
man, erred in denying to man any knowledge of the Infinite. Between the 
false Hamiltonian philosophy and Absolutism lies that true philosophy which 
yields to man some positive knowledge of the Infinite or Absolute. N. B. — In 
parallelism alone has man acquired a complete positive knowledge of an In- 
finite. 

Schol. — The laws of geometrical equality being eternal — not the result of 
power, such as the physical laws of the universe — therefore the why of every 
geometrical law seems possible to man. Besides, "Is not the conception of 
space common to all men, and does it not contain in it all that is necessary 
for a science of Geometry ? If so, there should be nothing disputable or open 
to cavil in the minds of competent thinkers. "| 

The student should distinguish between theoretical and actual Infinites : 
thus, asymptotes and numbers are infinite only in theory, &c, &c. 

*Lond. Edin. and Dub. Phil, Mag., 1857, Vol. XIII. page 413, as quoted from "Nichol's Cyclo- 
paedia of the Physical Sciences. " 
fPhilosophy of the Unconditioned. (See Hamilton's Discussions.) 
+Dr. Day on Parallels. Lond. Edin. and Dub. Phil. Mag., 1S57, Vol. XIII, page 156. 



c <c 



: 

■ 

OCT 

_*c 



BE 



cc 



c <c 



cc 
<^cc 
cc 

cc 
xc 
cc 
^lC 

xc 

JCC < 

= cc 
cc 







: 


« 


c 


4 


c 


1 




— 



^cc 
-^c<r 

cc 

r<c , 

cc ^ 









_ c <c 

crc 



& dC 



<r c 



crc 



dcrd 



_-- c 



_CC i 

53^ CT 



c 

^ cc 
J cc 
— cc 
__ cc 
- <r<z 

— cc 

_ c<^ 

Cc 

cc 

cc 



<^ c 

<£ c 



-CC 

CLCc 



<scc 



Cc OCT 
'<:< c 












OS c 



Cc 

cc 






Cr c<^ <^ C 



^"C^ 



























':-C c; 



>-_ C^ o 



<T "c- 



c 




5 cr 

55 c ^ 


r ^j| 






IS 


^^-C 




-— - • '-^ CZ <T~ 


f ^K f 




c 


===== c c: 


CF 


«JF 




«... ccc 




<fc^V 


~< 


: C.C'CT" 


wd~~< 


&C <; 




S2c<S 

JCxc"« 


C c 


E 


5£^ 


4 




i< cr. 


^s 


X-V. 






c 









; - egg ; 



LIBRARY OF CONGRESS 



II lilil Hi UuiM urn 
003 646 314 4 



